Delamination starting from a stress free edge is the main mode of failure of cross-ply laminates The analysis of singularities and resulting stress concentrations at interfaces along these edges are not sufficient to provide a satisfying model of onset mechanisms In particular, the model is by far less sensitive to the layers thickness than observed in traction experiments A refined asymptotic process is proposed which takes into account the existence of surface flaws (micro-cracks or notches) Numerical results prove that this model is much more in agreement with the experiments Moreover, the knowledge of the interface toughness, in addition with the experiments, allows to estimate a characteristic fracture length, a kind of process zone It is a material property and in a first approximation it arises to be of the same order of magnitude than the characteristic inhomogeneity length of the components, ie the fibers diameter

A method based on singularity theory to predict edge delamination of laminates

A survey is given of transonic small disturbance theory. Basic equations, shock relations, similarity laves, lift and drag integrals are derived., The airfoil boundary value problem is formulated. Finite difference methods and computational algorithms are described. Results are compared with other calculation methods and experiments.

Modern Developments in Transonic Flow

In two- and three-dimensional linear elasticity, the singularities together with matched asymptotic expansions allow to extend the brittle fracture mechanics. Although there exist some difference between 2D and 3D approaches, the usual crack tip singularity exponent \(\frac{1}{2}\) remains in both cases the hinge value between strong and weak singularities. In 3D, 0 was expected to be also a hinge, but it seems difficult to exhibit solutions with a negative exponent. One aim of this paper is to investigate numerically such specific cases and to derive some asymptotics of the classical stress intensity factors. The second part is dedicated to prove that, in case of a small linear ligament, negative exponents cannot exist.

On 3D Cracks Intersecting a Free Surface in Laminated Composites

The effect of throughflow and magnetic field on the onset of Benard convection in a horizontal layer of micropolar fluid permeated between two rigid, isothermal and micro-rotation free boundaries is studied. The determination of the critical Rayleigh number entails in solving the eigenvalue problem numerically for which the single-term Galerkin method is employed. It is established that both stabilizing and destabilizing factors can be enhanced by throughflow.

Effect of throughflow and magnetic field on Bénard convection in micropolar fluids

We present an asymptotic and numerical study of a class of junctions between rods and bulk bodies in the framework of two-dimensional elasticity. The junctions under consideration are such that the depth of the embedding of the rod into the bulk is small: it is zero or of the same order than the thickness of the rod. As a result, the junction is modelized by a flexible clamping. The study, mainly devoted to the elastic displacements and stress fields within the bulk, is performed using the matched asymptotic expansions method with different expansions in the vicinity of the junction and far from it. Some terms of these expansions involve elasticity problems with point forces and moments, which are analyzed by the singularity theory. Numerical examples are given.

Approximation of a two dimensional problem of junctions

Asymptotic methods in fluid mechanics

When handling experimental data received as far back as 1944–1948 Halperin, Gorsky, Kovalev and Christianovitch have discovered one of the main property of transonic flows, which they formulated as a stabilization law[1]. This law is confirmed by numerous wind tunnel tests, which were performed at different times in different countries. The most complete discussion of this experiments was given by Holder in 1964 in a lecture[2] commemorating the names of Reynolds and Prandtl.

Stabilization Law and Drag in Transonic Range of Velocities

On the three-dimensional sonic flow of an ideal gas past a body

THE paper examines the sets of natural functions and eigenfunctions introduced by Guderley for the study of transonic flow round finite bodies by an ideal gas, in the context of Karman's equations. We shall be examining uniform gas flow at a speed close to the velocity of sound round the profile of a wing of infinite span or a solid of revolution. The part played by dissipative processes is assumed to be negligibly small, so that the entire investigation can be based on Karman's approximate equation for the potential of the disturbances. It is assumed that the solution is known for a flow sonic at infinity. The solution for a flow whose speed deviates from the speed of sound in the undisturbed state, can be written as the sum of two terms. The first gives the asymptotic laws for the damping of the disturbances in the flow with critical velocitv, while the second enables account to be taken of the deviation of the velocity at infinity from the speed of sound. A linear equation is obtained for the correction potential, and its solution is sought as an infinite series in functions which depend on one similarity variable. These functions are called natural if the particle velocity components defined by them have no singularities on the limiting characteristic of the sonic flow. As a rule, the solutions required in the theory of transonic flow may be expanded in terms of these functions. The fundamental fact is established that the natural functions are expressible via the solutions of the eigenvalue problem for an ordinary second-order differential equation, reduced to the self-conjugate form. Series in the natural functions can be expressed by means of solutions of the eigenvalue problem, thus simplifying the investigation of the completeness of the system generated by them. Finally, extremely simple expressions are obtained, whereby the natural functions can be expressed in terms of polynomials.

On uniform flow at transonic speeds round finite bodies

A mathematical foundation is presented for the law of gas-parameter stabilization during near-sonic flow of a stream around bodies, which is known in experimental aerodynamics. The quantitative formulation of this law relies on a simple asymptotic analysis of the solutions of the Karman equations. The numerical computation performed for the velocity field around a body of revolution whose meridian section is a Chaplygin profile confirmed the deductions of asymptotic theory to high accuracy.

V. N. Diesperov

Papers

Asymptotic methods in fluid mechanics

Stabilization Law and Drag in Transonic Range of Velocities

On the three-dimensional sonic flow of an ideal gas past a body

On uniform flow at transonic speeds round finite bodies

Stabilization law for transonic flows around bodies of revolution